A finite element method is known as an approach to analyze the deformation of an object. In the finite element method, an object is approximated for representation into a set of a plurality of elements, and the deformation of the object is analyzed by use of the set of plurality of elements. More specifically, according to one known example of the finite element method, the shape of an object to be analyzed is approximately represented by means of elements, and then the deformation of the object is analyzed with the use of the elements. Each of the elements (plane elements or plane-like elements) may be defined to be a membrane element, a shell element having a thin wall, an element having a quadrilateral plane, an element having a triangular plane, etc.
Approximation accuracy of the finite element method depends on the type of a selected technique to partition an object into a plurality of elements, as hereinafter referred to as “element generation” or “mesh generation.” Therefore, in order to reduce errors in results of analysis by the finite element method (hereinafter, referred to as “analysis error”), an object to be analyzed is partitioned into a plurality of elements, such that the size or the shape of each element is proper with respect to the position of each element within the object, for example, such as disclosed in JP Hei 04-268674 and JP Hei 11-25292.
JP Hei 04-268674 discloses techniques of, once an analyzed object, i.e., a structure has been partitioned into a finite number of meshes (elements), optimizing the mesh generation, by moving nodes which have been initially assigned to the analyzed object, in consideration of the resulting errors in analysis results of such as stress, strain energy, etc. This technique provides one example of modification of elements in shape.
JP Hei 11-25292 discloses techniques of generating meshes of an analyzed object, each of which has the size meeting analysis accuracy required, and techniques of selecting the type of mesh generation as one of uniform mesh generation, non-uniform mesh generation, and junction mesh generation, depending upon the location of each mesh within the analyzed object. This technique provides another example of modification of elements in shape.
One example of a conventional mesh generation technique exists that allows the surface of an object to be partitioned into a plurality of meshes or elements so that every mesh or element may be desirable in shape. The conventional technique enables mesh generation to be performed so that every one of a plurality of elements (quadrilateral elements) into which the surface of an object is partitioned may be formed to be as close as possible to a square, i.e., a quadrilateral not distorted.
Further, in the above conventional technique, for evaluation to determine whether the shape of each element generated by mesh generation is desirable or not, the geometric distortion of the shape of each element due to the mesh generation is expressed by means of four geometric characteristic quantities, namely, the aspect ratio; the amount of warping; the amount of skew; and the amount of trapezoid, of each element.
The aspect ratio is defined, for example, as the ratio between a first and a second distance. The first distance is measured between a reference point of a distorted element, which is generally located at the center of the distorted element, and the midpoint of a first one of four sides of the distorted element (hereinafter, referred to as “element-sides”). The second distance is measured between the reference point, and the midpoint of a second one of the four element-sides, which is a selected one of two element-sides consecutive to the first element-side.
The amount of warping is defined, for example, with the use of a projection plane which is assumed for a distorted element such that the projection plane is equidistant from four nodes of the distorted element. More specifically, the amount of warping is defined as the arc sine of the ratio of a projection height equal to the distance between the projection plane and each node of the distorted element, to the half of the length of a selected one of the four element-sides of the distorted element.
The amount of skew is defined, for example, as the difference 90 degrees minus a smaller one of angles between a first and a second line. The first line passes through both the aforementioned reference point and the midpoint of the aforementioned first element-side. The second line passes through both the reference point and the midpoint of the aforementioned second element-side.
The mount of trapezoid is defined, for example, using a triangle for each element-side of a distorted element. The triangle is formed by a combination of the present element-side, and two rectilinear segments connecting the aforementioned reference point and each of the two nodes of the present element-side. In the example, the amount of trapezoid is then defined as four times the smallest one of four areas of four triangles which have resultantly generated for the four element-sides, respectively, divided by the total of the four areas.
Then, according to the above-described conventional technique, the actual value of each of the aforementioned four geometric characteristic quantities is compared with an allowable value, per each element. If the actual value of at least one of the four geometric characteristic quantities, for an element, exceeds the allowable value, then it is determined that the element is not proper in shape.